In macroscopic system of classical mechanics the motion of a particle is governed by Newton's second law :
.......(1)
Where F is force on the particle, m is it's mass and, t is the time, a is the acceleration given by a= dʋ/dt= (d/dt) (dx/dt)= d²x/dt², where ʋ is the velocity .Equation (1) contains the second derivatives of coordinate x with respect to time. To solve this, we must carry out 2 integrations. This introduces two arbitrary of constant c1 and c2 into the solution, and
x= g(t, c1, c2).... (2)
Where g is some functions of time. We must posses at a given time t₀ to be able to predict the future motion of the particle and if we know what at t₀ the particle is at point x₀, we have,
x₀= ɡ(t₀,c₁,c₂).....(3)
Since we have two constants to determine, and more information is needed. Differentiating.. (2) we have
If we also known that time t₀ the particle has velocity ʋ₀, then we have the addition of a relation,
........... (4)
We may use equation (3) and (4) to solve for c₁ and c₂ in terms of ʋ₀and x₀ known c₁ and c₂ and equation (2) to predict the exact future motion of the particles.
As an example of equation (1) and (4) to consider the vertical motion of the particles and it's Earth's gravitational field. The x - axis point is upwards. The force on the particles is downward and is F=-mg, where 'g' is the gravitational acceleration constant value. According to Newton's second law equation (1) is - mg= md²x/ dt², so d²x/dt²= - g. A single integration giving dx/dt =- gt + c₁.The arbitrary constant is c₁ can be found if we know that time t₀ the particle had position x₀. We can predictions the future position of the particles.
The classical mechanics, potential energy is V of particles moving in to one dimensional is defined, ...... ..
.......... (5)
For examples, for a particles is moving in the Earth's gravitational field, ðV/ðx =− F= mɡ and intevration gives V= mgx + c, where c is an arbitrary constant Vale. We are free to set that zero level of potential energy, we can choose c= 0, we have V = mgx as the potential energy functions.
The classical mechanics means a specific of the velocity and position of the each particles of the system at the some instant of time. Plus the specific of the forces acts on the particles. According to Newton's 2nd law, gives the state of a system at any time, it's future motion and states are exactly determined. Equation (2) and (4). The impression success of Newton's law in6explain, planetary motion and many philosopher's to use Newton's laws an argument for all philosophical determination. The mathematician and astronomer Laplace, assumed that the universe consisted of nothing but particles obeyed Newton's laws. Therefore, given the statement of the universe at the some instant. The future energy and motion of everything in the universe was completely determined. A super being able to known state of the universe at any instant could in this principle, calculated the all future motions.
Above given exact knowledge of the present state of classical mechanical systems and we can predict it's future state. However, the Heisenberg uncertainty principle shows that we can't determining simultaneously the exact velocity and position of the microscope objects of particles, so that very knowledge required by classical mechanics for predicted the future motion of a system can't be obtained. We must be the constant in quantum mechanics and with something less than complete predictable of the exact future motion.
Quantum mechanics will be a postulated the basic principles and then use postulates to deducing experimentally testable consequences such as the energy levels of the atom. The description of the state of system in quantum mechanics, we can postulates the existence of a function ψ of the particles and it's coordinates called as the, wave function or state function. The state is generally, changes with time, ψ is also wave function of time. For one - particles in one- dimensional system we have ψ= ψ (x, t). The wave function's contains all possible information about the system, so instead of " The state described by the wave function ψ" and we simply say " the state ψ" Newton's second laws telling us how to find the future state of the classical mechanics system from knowledge of it's present state, and we want an equation that tells us how that wave function changing with time. For a one - particle, one-dimensional systems, this equation is postulates to be
........ (6)
Where the constant h(h-bar) is defined as
ħ≡ h/2π...........(7)
The concept of wave equation and function governing it's change with time and it's discovered in 1926's by Austrian Scientists Erwin Schrodinger (1887-1961) in this equation, as the time- dependent equation or schrodinger wave equation i=
√ -1, m is the mass of particles and V (x, t) is the potential and energy function of the system.
The time - dependent schrodinger equation contains the 1st derivatives of the wave function with respect to times and allowing us to calculate the future wave state at any time, wave function at time denoted t₀.
The total wave function contains all the information and we can possibly knows about this system described. The wave function ψ gives us about the results of the measurement of the x coordinate of the particles, but we can't expect ψ involves the definite specification of positions that the state of a classical mechanics does. The exact correct answer to this question was provided by max born then schrodinger discovered that equation. But Born postulates that for a one-particle and one- dimensional system,
∣ ψ(x, t) ∣ ² dx...... (8)
Giving the probability at time is 't' of finding the particles in the region of the x axis and lying between x and x + dx (8) the bars denoted the absolute value and dx is infinitesimal length on the x axis. The wave function ∣ ψ (x, t) ∣ ² is the probability density for finding the particles at various place on the x axis. For example, suppose that at the particular time t₀ the particle is a state characterized by the wave function ae⁻ᵇˣ², where a and b are real constants. If we measure the particles position at time t₀, we get any value of x, because the probability density is a²e⁻ᵇˣ² is the nonzero at everywhere. Values of x in the region around x=0 are more likely to be found than other values, but since lψl² is a maximum origin in this case.
To relate the lψl² to the experimental of measures, we can take at many identical non interacting systems, each of which is in the same state ψ. The the particles position in each system is measured. If we had n system and made n measurement, and if dnₓ denotes the number of measurements for which we found the particle between x and x+ dx, then dnₓ/n is the maximum probability for finding particles between x and x+ dx. Thus
dnₓ/n= lψl² dx
and the graph of (1 /n) dnₓ/ dx versus x giving the probability density is lψl² as a function by taking of x. It might be that we could finding the probability - density function by taking for one system that was in the state ψ and repeated measuring the particle's position. This procedure is a wrong because the process of measurement actually changes the state of the system.
Quantum mechanics is statistically nature, and knowing that state, we can't predicted the result of the measurement with the certainty. But we can only predict the probability of the various results. The Bohr theories of the hydrogen atom specified the precision path of the electron and it's therefore not a correct quantum - mechanical pictures.
Quantum mechanics, doesn't say about that electron is distributed over a large region of space as a wave distributed. But it's rather probability pattern or wave function using to describe the electronic motion that behave a like waves and satisfy a wave equation.
The postulated of thermodynamics, first, second and third laws thermodynamics are stated in terms of macroscopic experience and hence are fairly and readily understood. The postulated of quantum mechanics state in terms of the microscopic world and appears quite abstract. But we should not fully understand the postulated of the quantum mechanism at first reading. As we can treat various examples, understanding of the postulated is will increase.
The schrodinger equation without any attempt to proving it's plausibility. By using analogies between geometrical optics the approximation to optics, when the wavelength of the light is much lesser than the size of the apparatus. But likewise, classical mechanics is approximation to wave mechanics from the classical mechanics based on the known relationship between the equation of geometrical and wave optics. But since many chemistry scientist are not particularly familiar with optics, this argument has been omitted. In any case such analogies only make Schrodinger equation seems as plausible. They can't be used to derive or prove this equation. The Schrodinger equation is a postulated of the theory to be tested by agreement of it's prediction with experiment.
Quantum mechanics providing that laws of motion for the microscopic particles. But experimentally, macroscopic objects obeying the classical mechanics as we making transition from microscopic to macroscopic particles. Quantum effects are associated with the de Broglie wavelength of macroscopic objects is essentially zeros. But in the limit λ→0, we can expect the time - depending Schrodinger equation to reduce to Newton's second law.
But similar situation is holding in the relation between special relatives reduces to classical mechanics. In the limit ʋ/c →0, where c is the speed lights and special relativity reduces to classical mechanics. The form of the quantum mechanics have not been achieved.
By, Historically quantum mechanics is first formulated in 1925 by noble winner Heisenberg, jordan and Born using the matrics before several months Schrodinger 1926, formulation used diffential equations. But Schrodinger proved that the Heisenberg formulation. But in 1926, Jordan and Dirac, working independently and given, formulated quantum mechanics in abstracts version, it's called " transformation theory" that's generalization of matrix mechanics and wave mechanics. In 1948, Feyman devised the correct path integral formulation of the quantum mechanical research.
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